Table Of ContentBIRTH OF LIMIT CYCLES FOR A CLASS OF
CONTINUOUS AND DISCONTINUOUS
DIFFERENTIAL SYSTEMS IN (d+2)–DIMENSION
5
1
0
JAUME LLIBRE1, MARCO A. TEIXEIRA2 AND IRIS O. ZELI2
2
n
a Abstract. The orbits of the reversible differential system x˙ =
J −y, y˙ = x, z˙ = 0, with x,y ∈ R and z ∈ Rd, are periodic with
5 the exception of the equilibrium points (0,0,z). We compute the
1
maximumnumberoflimitcycleswhichbifurcatefromtheperiodic
orbits of the system x˙ = −y, y˙ = x, z˙ = 0, using the averaging
]
S theory of first order, when this system is perturbed, first inside
D the class of all polynomial differential systems of degree n, and
. second inside the class of all discontinuous piecewise polynomial
h
differential systems of degree n with two pieces, one in y > 0 and
t
a
the other in y < 0. In the first case this maximum number is
m
nd(n−1)/2, and in the second is nd+1.
[
2
v
7
1. Introduction and statements of the main results
8
9
Limit cycles have been used to model the behavior of many real pro-
1
0 cess and different modern devices. In general to prove the existence
.
1 of limit cycles is a very difficult problem. One way to produce limit
0 cycles is perturbing differential systems that have a linear center. In
5
this case, the limit cycles in a perturbed system bifurcate from the pe-
1
: riodic orbits of the unperturbed center. The search for the maximum
v
i number of limit cycles that polynomial differential systems of a given
X
degree can have is part of 16th Hilbert’s Problem and many contribu-
r
a tions have been made in this direction, see for instance [11, 13, 16] and
the references quoted therein.
Recently the theory of limit cycles has also been studied in discon-
tinuous piecewise differential systems. The analysis of these systems
can be traced from Andronov et al. [1] and still continues to receive
attention by researchers. Discontinuous piecewise differential systems
is a subject that have been developed very fast due to its strong appli-
cations to other branches of science. Currently such systems are one of
2010 Mathematics Subject Classification. 34A36, 34C25, 34C29, 37G15.
Key words and phrases. limit cycle, averaging method, periodic orbit, polyno-
mial differential system, discontinuous polynomial differential systems.
1
2 J. LLIBRE, M.A. TEIXEIRA ANDI.O. ZELI
the connections between mathematics, physics and engineering. These
systems model several phenomena in control systems, impact in me-
chanical systems, nonlinear oscillations and economics see for instance
[2, 3, 4, 6, 14, 28]. Recently they have been shown to be also relevant as
idealized models for biology [15] and models of cell activity [7, 32, 33].
For more details see Teixeira [31] and all references therein.
As we have said it is not simple to determine the existence of limit
cycles in a differential system. The simplest case for determining limit
cycles is in planar continuous piecewise linear systems when they have
only two linear differential systems separated by a straight line. Even
in this simple case, only after a delicate analysis it was possible to show
the existence of at most of one limit cycle for such systems, see [8] or
an easier proof in [20].
Planar discontinuous piecewise linear differential systems with only
two linearity regions separated by a straight line have been studied re-
cently in [10, 12], among other papers. In [10] some results about the
existence of two limit cycles appeared, so that the authors conjectured
that the maximum number of limit cycles for this class of piecewise
linear differential systems is exactly two. However in [12] strong nu-
merical evidence about the existence of three limit cycles was obtained.
As far we know the example in [12] represents the first discontinuous
piecewise linear differential system with two zones with 3 limit cycles
surrounding a unique equilibrium. Recently in [22] it is proved that
such a system really has three limit cycles.
There are several papers studying the limit cycles of the continuous
piecewise linear differential systems inR3, seeforinstance [5,21,23,24,
25]. Our goal is study the periodic solutions of discontinuous piecewise
polynomial differential systems in Rd+2. More precisely the objective
of this paper is to study the existence of limit cycles in continuous
and discontinuous piecewise polynomial differential systems in Rd+2,
where the discontinuous differential system has two zones of continuity
separated by a hyperplane. Without loss of generality we shall assume
that the set of discontinuity is the hyperplane y = 0 in Rd+2. So we
consider the linear differential system in Rd+2 given by
x˙ =−y
(1) y˙ = x
z˙ = 0
ℓ
for ℓ = 1,...,d and x,y ∈ R, z ∈ Rd, where the dot denotes de-
rivative with respect to the time t, which is reversible with respect to
BIRTH OF LIMIT CYCLES 3
φ(x,y,z) = (x,−y,z) . First we are interested in studying the exis-
tence of limit cycles of the continuous polynomial differential system
given by
x˙ =−y +εP (x,y,z),
a
(2) y˙ = x+εP (x,y,z),
b
z˙ = εP (x,y,z),
ℓ cℓ
andafter we also study the existence of limit cycles ofthe discontinu-
ous piecewise polynomial differential system formed by two polynomial
differential systems separated by the hyperplane y = 0, namely
x˙ = −y +εP (x,y,z),
a
y˙ = x+εP (x,y,z), if y > 0
b
z˙ = εP (x,y,z),
ℓ cℓ
(3)
x˙ = −y +εQ (x,y,z),
a
y˙ = x+εQ (x,y,z), if y < 0
b
z˙ = εQ (x,y,z),
ℓ cℓ
where ε 6= 0 is a small parameter and ℓ = 1,...,d. In this systems the
polynomials P , P , P , Q , Q , Q are of degree n in the variables
a b cℓ α β γℓ
x, y and z, more precisely
n n
P (x,y,z) = a xiyjzk, P (x,y,z) = b xiyjzk,
a ijk b ijk
i+j+k=0 i+j+k=0
X X
n n
P (x,y,z) = c xiyjzk, Q (x,y,z) = α xiyjzk,
cℓ ℓijk a ijk
i+j+k=0 i+j+k=0
X X
n n
Q (x,y,z) = β xiyjzk, Q (x,y,z) = γ xiyjzk.
β ijk γℓ ℓijk
i+j+k=0 i+j+k=0
X X
In this expressions k is a multi-index and i+j+k denotes i+j+k +
1
...+k , zk denotes the product zk1...zkd with z = (z ,...,z ) ∈ Rd,
d 1 d 1 d
and a denotes the coefficient a of xiyjzk1 ...zkd.
ijk ijk1...kd 1 d
It is clear that systems (2) and (3) coincide for ε = 0 and they have
linear centers at every plane z = constant. In this paper we establish
for ε 6= 0 sufficiently small the maximum number of limit cycle of these
systems that bifurcate from the periodic orbits of these linear centers
using the averaging theory of first order. The following result presents
the results for the continuous case.
4 J. LLIBRE, M.A. TEIXEIRA ANDI.O. ZELI
Theorem 1. Using the averaging theory of first order for |ε| =6 0 suffi-
ciently small the maximum number of limit cycles of the polynomial dif-
ferential system (2) is at most nd(n−1)/2, and this number is reached.
Inthenexttheoremwepresentresultsforthediscontinuouspiecewise
polynomial differential system (3).
Theorem 2. Using the averaging theory of first order for |ε| =6 0 suffi-
ciently small the maximum number of limit cycles of the discontinuous
piecewise polynomial differential system (3) is at most nd+1, and this
number is reached.
Corollary 3. Under the assumptions of Theorem 1 if additionally
a = b = 0 for all k, the limit cycles can be chosen as close to
00k 00k
the origin of Rd+2 as we want.
Corollary 4. Under the assumptions of Theorem 2 if additionally
a = b = α = β = 0 for all k, the number of limit cycles
00k 00k 00k 00k
of the discontinuous piecewise polynomial differential system (3) is at
most nd(n − 1), and this number is reached. Furthermore, the limit
cycles can be chosen as close to the origin of Rd+2 as we want.
Corollaries 3 and 4 provides information of the Hopf bifurcation of
systems (2) and (3). More precisely, Corollaries 3 and 4 show that at
least nd(n−1)/2 and nd(n−1) limit cycles of systems (2) and (3) can
bifurcate from the origin of Rd+2, respectively. The results of Corollary
3 in the particular case n = 2 coincides with the result obtained in
Theorem 1 of [26].
To prove these results we use the classical averaging theory, see for
instance [30, 34] for a general introduction to this subject. This theory
have been used for years to deal with continuous differential systems.
The principle of averaging has been extended in many directions and
recently in [19] the authors extend the averaging theory for detecting
limit cycles of certain discontinuous piecewise differential systems, via
the Brouwer degree and the regularization theory.
As far as we know this method is one of the best methods for deter-
mining limit cycles in discontinuous piecewise differential systems and
has already been used by some authors. In [17] the method is used
for determining the maximum number of limit cycles that bifurcate
from the periodic solutions of some family of isochronous cubic poly-
nomial centers perturbed by discontinuous piecewise cubic polynomial
differential systems with two zones separated by a straight line. In [18]
limit cycles for discontinuous piecewise quadratic differential systems
with two zones was studied using the averaging theory. Also in [29]
BIRTH OF LIMIT CYCLES 5
the averaging theory was applied to provide sufficient conditions for
the existence of limit cycles of discontinuous perturbed planar centers
when the discontinuity set is a union of regular curves.
We have organized this paper as follows. In Section 2 we briefly
present notation and basic concepts of the averaging theory of first or-
der for continuous differential systems (see Theorem 5) and for discon-
tinuous differential systems (see Theorem 6). In Section 3 we present
the proof of results of this paper. More precisely, the proof of Theorem
1 is presented in the Subsection 3.1, and in the Subsection 3.2 we prove
the Theorem 2. Finally in the Subsection 3.3 we presented the proofs
of the Corollaries 3 and 4.
2. Basic results in averaging theory
In this section we present the basic results from the averaging theory
of first order that we shall use for proving the results of this paper.
The following theorem provides a method for studying the existence
of periodic orbits of a differential system. For more details on the
averaging method see for instance [34].
Let D be an open subset of Rn. We denote the points of R×D as
(t,x), and we take the variable t as the time.
Theorem 5. Consider the differential system
(4) x˙ = εF(t,x)+ε2R(t,x,ε), x(0) = 0,
where F : R×D → Rn and R : R×U ×(−ε,ε) → Rn are continuous
functions and T−periodic in the first variable. Define the averaging
function f : D → Rn as
T
(5) f(x) = F(s,x) ds,
Z0
and assume that
(i) the functions F, R, D F, D2F and D R are defined, con-
x x x
tinuous and bounded by a constant M (independent of ε) in
[0,∞)×D and for ε ∈ (0,ε ],
0
(ii) for p ∈ D with f(p) = 0 we have |J (p)| =6 0, where |J (p)|
f f
denotes the determinant of the Jacobian matrix of f evaluated
at p.
Then for |ε| > 0 sufficiently small there exists a T−periodic solution
x(t,ε) of the system (4) such that x(0,ε) → p as ε → 0.
Nowleth : R×D → RbeaC1 functionwith0 ∈ Rasaregularvalue,
and Σ = h−1(0). Let X,Y : R × D → Rn be two continuous vector
6 J. LLIBRE, M.A. TEIXEIRA ANDI.O. ZELI
fields and assume that h,X and Y are T−periodic in the variable t.
We define a discontinuous piecewise differential system as
X(t,x) if h(t,x) > 0,
(6) x˙ = Z(t,x) =
Y(t,x) if h(t,x) < 0.
(cid:26)
We rewrite the discontinuous differential system as follows. Consider
the sign function defined in R\{0} as
1 if u > 0,
sign(u) =
−1 if u < 0.
(cid:26)
Then system (6) can be written as
x˙ = Z(t,x) = F (t,x)+sign(h(t,x))F (t,x),
1 2
where
1 1
F (t,x) = (X(t,x)+Y(t,x)) and F (t,x) = (X(t,x)−Y(t,x)).
1 2
2 2
The following theorem is a version of Theorem 5 for studying the
periodic solutions of discontinuous differential systems.
Theorem 6. Consider the discontinuous differential system
(7) x˙ = εF(t,x)+ε2R(t,x,ε),
with
F(t,x) = F (t,x)+sign(h(t,x))F (t,x),
1 2
R(t,x,ε) = R (t,x,ε)+sign(h(t,x))R (t,x,ε),
1 2
where F ,F : R × D → Rn, R ,R : R × D × (−ε,ε) → Rn and
1 2 1 2
h : R×D → R are continuous functions, T−periodic in the variable t.
We also suppose that h is a C1 function with 0 as a regular value and
we denote Σ = h−1(0). Define the averaged function f : D → Rn as
T
(8) f(x) = F(s,x) ds,
Z0
and assume that
(i) the functions F ,F ,R ,R and h are locally Lipschitz with re-
1 2 1 2
spect to x;
∂h
(ii) (t,x) 6= 0 for all (t,x) ∈ Σ;
∂t
(iii) for p ∈ C with f(p) = 0, there exist a neighborhood U ⊂ C of p
such that f(z) 6= 0 for all z ∈ U\{p} and d (f,U,0) 6= 0 (d
B B
is the Brouwer degree of f in p).
BIRTH OF LIMIT CYCLES 7
Then for |ε| > 0 sufficiently small there exists a T−periodic solutions
x(t,x) of system (7) such that x(t,ε) → p as ε → 0.
For a proof of Theorem 6 see Theorem A and Proposition 2 in
[19]. Here we emphasize that if f in (8) is C1 then the hypotheses
d (f,U,0) 6= 0 holds if |J (p)| =6 0, see for more details [27].
B f
3. Proof of the main results
We devoted this section for the proof of results of this paper. For
this we consider the notations introduced in the Introduction.
3.1. Proof of Theorem 1. Applying the change of variables
(9) x = rcosθ, y = rsinθ and z = z,
system (2) becomes
n
ε
θ˙ = 1+ ri+jzk a cosiθsinj+1θ+b cosi+1θsinjθ ,
ijk ijk
r
i+j+k=0
n X (cid:0) (cid:1)
r˙ = ε ri+jzk a cosi+1θsinjθ +b cosiθsinj+1θ ,
ijk ijk
i+j+k=0
Xn (cid:0) (cid:1)
z˙ = ε c ri+jzkcosiθsinjθ,
l lijk
i+j+k=0
X
Essentially for ε 6= 0 sufficiently small we study the existence of limit
cycles bifurcating from periodic orbits of this system when ε = 0, and
that are contained in the cylindrical annulus
(10) A˜ = {(θ,r,z) : r ≤ r ≤ r , θ ∈ S1, z ∈ Rd}.
0 1
So for ε sufficiently small θ˙ > 0 for every (θ,r,z) ∈ A˜.
˜
Now taken as new independent variable θ instead of t so (2) in A
can be written as
r′ = εF (θ,r,z)+O(ε2)
1
(11)
z′ = εF (θ,r,z)+O(ε2),
ℓ ℓ+1
8 J. LLIBRE, M.A. TEIXEIRA ANDI.O. ZELI
for ℓ = 1,...,d, where the prime denotes derivative with respect to
the variable θ, and
n
F (θ,r,z) = ri+jzk a cosi+1θsinjθ
1 ijk
i+j+k=0
X (cid:0)
(12) +b cosiθsinj+1θ ,
ijk
n
(cid:1)
F (θ,r,z) = ri+jzk c cosiθsinjθ,
ℓ+1 ℓijk
i+j+k=0
X
for ℓ = 1,2,...,d.
System (11) is 2π–periodic with respect to the independent variable
θ, and satisfies the hypotheses of the Theorem 5 for ε small and D
0
fixed. Now we compute the averaged function f = (f ,f ,...,f )
1 2 d+1
given in (5) with T = 2π.
Now taking
2π
µ = cospθsinqθ dθ,
(p,q)
Z0
note that µ 6= 0 if only if p and q are simultaneously even.
(p,q)
So we obtain
(13)
2π n
f (r,z) = F (θ,r,z) dθ = ri+jzk a µ
1 1 ijk (i+1,j)
Z0 i+jX+k =0
i odd, j even
n
+ ri+jzk b µ
ijk (i,j+1)
i+jX+k =0
i even, j odd
2π n
f (r,z) = F (θ,r,z) dθ = ri+jzk c µ ,
ℓ+1 ℓ+1 ℓijk (i,j)
Z0 i+jX+k =0
i,j even
for ℓ = 1,2,...,d.
We split the proof in two parts. First assume n is odd, so
f (r,z) = A r +A r3 +...+A rn
1 1 3 n
BIRTH OF LIMIT CYCLES 9
where
(14)
n−p
A = zk a µ + b µ ,
p ijk (i+1,j) ijk (i,j+1)
Xk=0 i+Xj =p i+Xj =p
i odd,j even i even,j odd
¯
for p = 1,2,...,n. So we write f = rf with
1 1
f¯(r,z) = A +A r2 +...+A rn−1.
1 1 3 n
¯
Since r > 0 it is sufficient to solve (f ,f ,...,f ) = (0,...,0) to
1 2 d+1
¯
determine the number of solutions of f ≡ 0. As f is a polynomial in
1
the variables r and z ∈ Rd of degree n−1 and f are polynomials in
l+1
the variables r and z ∈ Rd of degree n for l = 1,2,...,d, by B´ezout’s
theorem (see [9]) (f¯,f ,...,f ) has at most nd(n − 1) solutions.
1 2 d+1
¯
However f is even on variable r then we consider only solutions with
1
r > 0, in this case the maximum number of solutions of f ≡ 0 is
nd(n−1)/2.
Now we prove that this number is reached. For this, we exhibit a
particular case for which this occurs. Let a ,b 6= 0 and we take zero
ij0 ij0
¯
all the other a ,b , then f (r,z) is a real polynomial that does not
ijk ijk 1
depend of z ∈ Rd of degree n−1, that is
f¯ = A +A r2 +...+A rn−1
1 1 3 n
where
A = a µ + b µ .
p ij0 (i+1,j) ij0 (i,j+1)
i+Xj =p i+Xj =p
i odd, j even i even, j odd
On the other hand, we take c = 0 if i,j,k ,k ,k ,...,k 6= 0
ℓijk 1 l−1 l+1 d
for each ℓ = 1,2,...,d so that
n
f (r,z) = z kℓ (2π c ).
ℓ+1 ℓ ℓ00kℓ
kXℓ=0
Inthisparticularcase, wecantakea ,b andc inorderthatall
ij0 ij0 ℓ00kℓ
¯
coefficientsoff andf arelinearlyindependent forallℓ = 1,2,...,d.
1 ℓ+1
¯
So we can choose these coefficients in such a way that f has (n−1)/2
1
simple positive real roots and f has n simple real roots for each
ℓ+1
ℓ = 1,2,...,d. Then (f¯,f ,...,f ) has nd(n−1)/2 solutions with
1 2 d+1
r > 0.
10 J. LLIBRE, M.A. TEIXEIRA ANDI.O. ZELI
¯
Now assume n is even. Then f = rf where
1
f¯(r,z) = A +A r2 +...+A rn−2.
1 1 3 n−1
with A given in (14).
p
¯
Note that f has degree n−1 as a polynomial in the variables r and
1
z and f given in (13) has degree n as polynomials in the variables r
ℓ+1
andz foralll = 1,2,...,d,so(f¯,f ,...,f )haveatmostnd(n−1)/2
1 2 d+1
solutions of type (r,z) with r > 0.
To prove that this number is reached we consider a ,b 6= 0 and
10k1 01k1
we take zero all the other a ,b . Then
ijk ijk
n−1
f¯(r,z) = A = zk1 a µ +b µ
1 1 1 10k1 (2,0) 01k1 (0,2)
kX1=0 (cid:0) (cid:1)
is a complete polynomial on variable z of degree n − 1. Now for
1
ℓ = 2,3...,d we take c ,c 6= 0 and we take zero all the other
1ij0 ℓ00kℓ
c so that
ℓijk
n
f (r,z) = ri+j c µ ,
2 1ij0 (i,j)
i+Xj =0
i,j even
n
f (r,z) = zkℓ c 2π,
ℓ+1 ℓ ℓ00kℓ
kXℓ=0
for ℓ = 2,3...,d.
Here we take a , b , c and c in such a way for that
10k1 01k1 1ij0 ℓ00kℓ
¯
all coefficients of f and f for ℓ = 1,2,...,d, are linearly inde-
1 ℓ+1
¯
pendent. Therefore we can choose these coefficients in order that f
1
has n−1 simple real roots, f has n/2 simple positive real roots and
2
f has n simple real roots for each ℓ = 2,3,...,d . In this case
ℓ+1
(f¯,f ,...,f ) = (0,...,0) has nd(n − 1)/2 solutions with r > 0.
1 2 d+1
Furthermore by independence of the coefficients these solutions can be
taken in a way that the Jacobian of f in all these solutions is nonzero.
This completes the proof of Theorem 1.
3.2. Proof of Theorem 2. Analogously to what we did in the previ-
ous section we shall study the existence of limit cycles bifurcating of
periodic solutions of system (3) when ε = 0 which are contained in the
cylindrical annulus A˜ defined in (10).
In A˜we have for εsufficiently small θ˙ > 0 forall (θ,r,z) ∈ A˜. Taking
˜
θ as independent variable system (3) in A becomes
